Note No. 23, Part 1 from the series "Cosmobiology of Mirny": The gap that cannot be forgotten — mass gap as the limiting case of soft memory

The Millennium Problem of Yang–Mills requires proving the existence of a mass gap: a nonlinear gauge medium cannot relax to the vacuum continuously — between the ground state and the first excited state there stands an exact energy barrier Δ > 0 that cannot be erased by any smooth perturbation. This note compares this exact gap with the soft gap of synaptic memory: finite barriers and power-law decay of the trace, described by a superposition of exponentials with a distribution of relaxation times (the spectrum law, Note 16, with reference to Fusi, 2005). The central question is posed: what is the critical distribution of barriers at which a soft gap hardens into an exact one? It is shown that the exact gap is the limiting case of a spectrum with a degenerate (delta-like) distribution, and that the bridge between gauge theory and glassy dynamics is a question about the critical tail of the barrier distribution, rather than about solving either of these problems separately. Analogy, not isomorphism.